Re: Scott and George's Teaching Thread

On Aug 23, 12:58 pm, Scott <ToaTe...@xxxxxxxxx> wrote:
On Aug 23, 9:09 am, george <gree...@xxxxxxxxxx> wrote:

When you see "=" in set theory, you need to think of it as a macro.
B=C
MEANS (when C and B are sets)
Ax[ xeB <--> xeC ].
It means "B is a subset of C and C is a subset of B".
The "=" is an abbreviation for something phrased PURELY in terms of e
(since,
this being set theory, EVERYthing is phrased in terms of set
membership).

So to practise, the subset symbol, denoted as a sideways "u" is also a
macro:
B subset C
means, when B and C are sets
Ax[ xeB -> xeC].

Right, 'subset of' (which is an English rendering of the subset
symbol) is a DEFINED 2-place predicate symbol of the language. The
definition:

Axy(x subset of y <-> Az(zex -> zey)).

We add that formula to the axioms of our set theory, as that formula
then is a definitional axiom. (Usually though, we leave off the
initial universal quantifiers, such as 'Axy', in definitions as such
quantification is understood to be implicit.)

But even more basically, the way george is setting it up (which
happens to be a fine way of setting up), even '=' is a defined 2-place
predicate symbol of the language.

First we could use this axiom:

x=y -> Az(xez <-> yez)

Then we add a definition:

x=y <-> Az(zex <-> zey).

And from that definition along with the definition of 'subset of', we
get:

x=y <-> (x subset of y & y subset of x).

And that gives us a very common way of proving that two sets x and y
are equal: Show that x is a subset of y and then that y is a subset of
x.

And, from the definition of '=' we get the ordinary axiom of
extensionality too:

Az(zex <-> zey) -> x=y

Also, from the definition of '=', we get:

x=x

And, from what we have so far, by induction on formulas we get a
formalization of Euclid's "equals for equals" principle (also along
the lines of a formalization of Leibniz's indiscernibility of
identicals principle), which roughly put is:

For all formulas P and all terms t and s,
t=s -> (P <-> P')
where P' is exactly like P except s occurs in P' in zero or more
places where t occurs in P (with the usual free-for restrictions).

Thus, as george mentioned, we have proven identity theory (as far as
the language of set theory goes) from our definition of '=' and our
set theoretic axioms (just the one I mentioned so far is all that is
needed for this particular purpose).

MoeBlee


.

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